Controllability Analysis of Fractional-Order Delay Differential Equations via Contraction Principle

This paper investigates the existence of solutions and the controllability for three distinct types of fractional-order delay differential equations, aiming to establish sufficient conditions for both existence and uniqueness while demonstrating controllability. Beginning with a fractional-order delayed system containing a nonzero control function, we apply the Banach fixed-point theorem to show that this system has a unique solution and satisfies the controllability property. Extending our analysis, we introduce an integral function with a delay term on the right-hand-side of the system, forming a more complex integro-fractional delay system. With a Lipschitz condition imposed on this newly introduced function, we establish the existence and uniqueness of solution, as well as the controllability of this system. In the final system, an integro-fractional hybrid model, an additional delayed function is embedded within the Caputo derivative operator, introducing distinct analytical challenges. Despite these complexities, we use the Banach fixed-point theorem and certain assumptions to demonstrate that the systems are controllable. Our approach is distinctive in incorporating delay functions on both sides of the related systems, which we support with theoretical results and illustrative examples. The paper outlines the fundamentals of fractional calculus, specifies the necessary assumptions, and uses fixed-point criteria to establish controllability with the existence of a solution, providing a clear framework for analyzing fractional-order control systems with delay functions.